Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the largest prime factor of non-zero Fourier coefficients of Hecke eigenforms

Sanoli Gun, Sunil L Naik

Abstract

Let $τ$ denote the Ramanujan tau function. One is interested in possible prime values of $τ$ function. Since $τ$ is multiplicative and $τ(n)$ is odd if and only if $n$ is an odd square, we only need to consider $τ(p^{2n})$ for primes $p$ and natural numbers $n \geq 1$. This is a rather delicate question. In this direction, we show that for any $ε> 0$ and integer $n \geq 1$, the largest prime factor of $τ(p^{2n})$, denoted by $P(τ(p^{2n}))$, satisfies $$ P(τ(p^{2n})) ~>~ (\log p)^{1/8}(\log\log p)^{3/8 -ε} $$ for almost all primes $p$. This improves a recent work of Bennett, Gherga, Patel and Siksek. Our results are also valid for any non-CM normalized Hecke eigenforms with integer Fourier coefficients.

On the largest prime factor of non-zero Fourier coefficients of Hecke eigenforms

Abstract

Let denote the Ramanujan tau function. One is interested in possible prime values of function. Since is multiplicative and is odd if and only if is an odd square, we only need to consider for primes and natural numbers . This is a rather delicate question. In this direction, we show that for any and integer , the largest prime factor of , denoted by , satisfies for almost all primes . This improves a recent work of Bennett, Gherga, Patel and Siksek. Our results are also valid for any non-CM normalized Hecke eigenforms with integer Fourier coefficients.
Paper Structure (12 sections, 18 theorems, 181 equations)

This paper contains 12 sections, 18 theorems, 181 equations.

Table of Contents

  1. Introduction and Statements of Results
  2. Preliminaries
  3. Symmetric $m$-th power of a Galois representation
  4. The Sato-Tate conjecture
  5. Trace of symmetric $n$-th power of matrices
  6. Cyclotomic polynomials and primitive divisors of $A^n-B^n$
  7. Largest prime factor of Fourier coefficients at prime powers
  8. Trace zero elements in the image of the symmetric $(q-1)$-th power of a Galois representation
  9. Proof of Theorem \ref{['thm1']}
  10. Proof of Remark \ref{['rmkthm2-1']}
  11. Proof of Corollary \ref{['Corpm']}
  12. Proof of Theorem \ref{['thm2']}

Key Result

Theorem 1

Let $n \geq 1$ be an integer and $\epsilon> 0$ be a real number. Then for almost all primes $p$, we have Further, if $q=P(2n+1)$ is sufficiently large, then the set of primes $p$ such that has positive lower density.

Theorems & Definitions (30)

  • Theorem 1
  • Remark 1.1
  • Corollary 2
  • Theorem 3
  • Remark 1.2
  • Theorem 4
  • Theorem 5
  • Lemma 6
  • Lemma 7
  • Theorem 8
  • ...and 20 more